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Results for “LIE GROUP”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Simultaneous Measurements of Noncommuting Observables: Positive Transformations and Instrumental Lie Groups

We formulate a general program for describing and analyzing continuous, differential weak, simultaneous measurements of noncommuting observables, which focuses on describing the measuring instrument autonomously, without states. The Kraus operators of such measuring processes are time-ordered products of fundamental differential positive transformations, which generate nonunitary transformation groups that we call instrumental Lie groups. The temporal evolution of the instrument is equivalent to the diffusion of a Kraus-operator distribution function, defined relative to the invariant measure of the instrumental Lie group. This diffusion can be analyzed using Wiener path integration, stochastic differential equations, or a Fokker-Planck-Kolmogorov equation. This way of considering instrument evolution we call the Instrument Manifold Program. We relate the Instrument Manifold Program to state-based stochastic master equations. We then explain how the Instrument Manifold Program can be used to describe instrument evolution in terms of a universal cover that we call the universal instrumental Lie group, which is independent not just of states, but also of Hilbert space. The universal instrument is generically infinite dimensional, in which case the instrument’s evolution is chaotic. Special simultaneous measurements have a finite-dimensional universal instrument, in which case the instrument is considered principal, and it can be analyzed within the differential geometry of the universal instrumental Lie group. Principal instruments belong at the foundation of quantum mechanics. We consider the three most fundamental examples: measurement of a single observable, position and momentum, and the three components of angular momentum. As these measurements are performed continuously, they converge to strong simultaneous measurements. For a single observable, this results in the standard decay of coherence between inequivalent irreducible representations. For the latter two cases, it leads to a collapse within each irreducible representation onto the classical or spherical phase space, with the phase space located at the boundary of these instrumental Lie groups.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

A Type II Hamiltonian Variational Principle and Adjoint Systems for Lie Groups

We present a novel Type II variational principle on the cotangent bundle of a Lie group which enforces Type II boundary conditions, i.e., fixed initial position and final momentum. In general, such Type II variational principles are only globally defined on vector spaces or locally defined on general manifolds; however, by left translation, we are able to define this variational principle globally on cotangent bundles of Lie groups. Type II boundary conditions are particularly important for adjoint sensitivity analysis, which is our motivating application. As such, we additionally discuss adjoint systems on Lie groups, their properties, and how they can be used to solve optimization problems subject to dynamics on Lie groups.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Lie group analysis of the point-reactor neutron kinetics equations for various reactivity models

This work applies Lie Group Theory to the nonlinear point-reactor neutron kinetics equations for several reactivity feedback and reactivity insertion models. The goal of this work is to connect known analytical solutions to their corresponding Lie groups as well as to obtain new analytical solutions and the circumstances from which they arise. The reactivity models we consider are the Nordheim-Fuchs Model, an arbitrary-order polynomial reactivity insertion model, and the Fuchs Ramp-Insertion Model. In all three models, we generalize the analysis when possible and obtain analytical solutions in every case. For the first two models, we succeed in finding the Lie groups corresponding to classical solutions and find additional solutions that were otherwise unknown. Finally, for the final model, we obtain a reduced-order nonlinear kinetics equation which is not analytically solvable, but by applying a second-order truncation, we find a novel closed-form solution and demonstrate its validity with a numerical example.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2018

Homomorphisms of Lie Groups

In this paper, the authors describe homomorphisms of Lie groups into the groups u(R) of invertible elements of rings R for a large class of rings R, which contains, in particular, subrings of matrix rings and also Noetherian rings.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Lectures on Lie Group Analysis: Solving Differential Equations Using Symmetries

These notes are meant to be a supplemental reference for the beginner Lie Group Analyst. It is assumed that the reader has a basic concept of the fundamentals of Lie Group Theory (LGT), e.g. has seen the derivation of the infinitesimal generator and understands the mathematical meaning behind invariance. An excellent reference is Albright et al., “Symmetry Analysis of Differential Equations: A Primer,”. The reader is urged to read at least the first three chapters of that document to be able to follow the outset of Chapter 2 of this document. The reader should also have a general understanding of calculus, ordinary differential equations, and partial differential equations.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1972

Control systems on Lie groups.

The controllability properties of systems which are described by an evolution equation in a Lie group are studied. The revelant Lie algebras induced by a right invariant system are singled out, and the basic properties of attainable sets are derived. The homogeneous case and the general case are studied, and results are interpreted in terms of controllability. Five examples are given.

Jurdjevic, V.↗
NASA NTRS1983

Some applications of Lie groups in astrodynamics

Differential equations that arise in astrodynamics are examined from the standpoint of Lie group theory. A summary of the Lie method is given for first degree differential equations. The Kepler problem in Hamiltonian form is treated by this method. Extension of the Lie method to optimal trajectories is outlined.

Jackson, A. A.↗
DOE OSTI2019

Lie groups of variable cross-section channel flow

This paper considers Lie groups of scaling transformations for the system of equations governing inviscid, compressible, quasi-one dimensional (quasi-1D) fluid flow in channels with variable cross-section. We determine the coupling between the admissibility of the group transformations and the equation of state (EOS) model used to close the system as well as the spatial and temporal dependence of the channel cross-section. The results presented extend prior group analysis performed in Ovsiannikov and Boyd et al. on the equations of gas dynamics for 1D flow. In the context of channel flow, the analysis in Ovsiannikov and Boyd et al. is only applicable for a very limited scope of channels, namely for channels possessing planar, cylindrical and spherical symmetry where the flow is truly 1D. To illustrate the extension achieved using the quasi-1D model, we consider the classical Noh problem set up in a channel with a variable area cross-section. First, a new set of canonical variables are derived using the admissible scaling transformations. Then, by re-expressing the governing equations of motion in terms of the new variables, a solvable system of ordinary differential equations (ODEs) is acquired. The resulting explicit solutions of the problem extend the previous results of Ramsey et al. to solutions of the Noh problem for quasi-1D flow.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2019

Birational Darboux Coordinates on Nilpotent Coadjoint Orbits of Classical Complex Lie Groups, the Case of 2 × 2 Jordan Blocks

We consider the problem of constructing birational Darboux coordinates on nilpotent coadjoint orbits of the complex Lie groups SO(N, ℂ) and Sp(N, ℂ). The nilpotent case is the most difficult one. Difficulties arise if the Jordan form of matrices from the orbit under consideration contains Jordan blocks of sizes of different parity. The desired coordinates have been found on orbits consisting of matrices with 1 × 1 and 2 × 2 Jordan blocks. Explicit formulas for them are given in the paper.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

Exact optical solitons in metamaterials with anti-cubic law of nonlinearity by Lie group method

In this paper, using Lie point symmetry analysis, we study the dynamics of the solitons for the nonlinear Schrodinger’s equation with anti-cubic nonlinearity. Some new doubly periodic solutions are obtained that degenerate to dark and bright soliton solutions. In our of best knowledge, the obtained solutions are new. Those obtained results have important applications in the understanding the nonlinear propagation theory of solitons in metamaterials.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1985

The general Lie group and similarity solutions for the one-dimensional Vlasov-Maxwell equations

The general Lie point transformation group and the associated reduced differential equations and similarity forms for the solutions are derived here for the coupled (nonlinear) Vlasov-Maxwell equations in one spatial dimension. The case of one species in a background is shown to admit a larger group than the multispecies case. Previous exact solutions are shown to be special cases of the above solutions, and many of the new solutions are found to constrain the form of the distribution function much more than, for example, the BGK solutions do. The individual generators of the Lie group are used to find the possible subgroups. Finally, a simple physical argument is given to show that the asymptotic solution for a one-species, one-dimensional plasma is one of the general similarity solutions.

Roberts, D.↗
DOE OSTI2023

Deep learning symmetries and their Lie groups, algebras, and subalgebras from first principles

Abstract We design a deep-learning algorithm for the discovery and identification of the continuous group of symmetries present in a labeled dataset. We use fully connected neural networks to model the symmetry transformations and the corresponding generators. The constructed loss functions ensure that the applied transformations are symmetries and the corresponding set of generators forms a closed (sub)algebra. Our procedure is validated with several examples illustrating different types of conserved quantities preserved by symmetry. In the process of deriving the full set of symmetries, we analyze the complete subgroup structure of the rotation groups SO (2), SO (3), and SO (4), and of the Lorentz group S O ( 1 , 3 ) . Other examples include squeeze mapping, piecewise discontinuous labels, and SO (10), demonstrating that our method is completely general, with many possible applications in physics and data science. Our study also opens the door for using a machine learning approach in the mathematical study of Lie groups and their properties.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1971

Similarity analysis of differential equations by Lie group.

Methods for transforming partial differential equations into forms more suitable for analysis and solution are investigated. The idea of Lie's infinitesimal contact transformation group is introduced to develop a systematic method which involves mostly algebraic manipulations. A thorough presentation of the application of this general method to the problem of similarity analysis in a broader sense - namely, the similarity between partial and ordinary differential equations, boundary value and initial value problems, and nonlinear and linear equations - is given with new and very general methods evolved for deriving the possible groups of transformations.

Na, T. Y.↗
DOE OSTI2018

Symmetry analysis of differential equations: a primer

Symmetry analysis, also known as group invariance, Lie-group, and similarity analysis, originated from the work of Lie [13], well over a century ago, in part to unify the large number of different techniques devised to solve specific differential equations. Lie’s approach to studying differential equations is based on three key ideas: symmetry, in variance, and Lie groups. It is assumed that the reader is unfamiliar with these concepts, and the purpose of this primer is to introduce them to the reader and illustrate their perhaps surprising applicability to the study of differential equations.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1987

Algebra and topology for applications to physics

The principal concepts of algebra and topology are examined with emphasis on applications to physics. In particular, attention is given to sets and mapping; topological spaces and continuous mapping; manifolds; and topological groups and Lie groups. The discussion also covers the tangential spaces of the differential manifolds, including Lie algebras, vector fields, and differential forms, properties of differential forms, mapping of tangential spaces, and integration of differential forms.

Rozhkov, S. S.↗